Growth of logarithmic derivative of meromorphic functions (Q2873582)

We use the standard notation of value distribution theory. Let NEWLINE\[NEWLINE\rho_p(f)=\limsup _{r\to \infty} \frac{\log _{p} T(r,f)}{\log r},\quad p \in {\mathbb N}, NEWLINE\]NEWLINE where \(\log_p\) denotes the \(p\)th iteration of the logarithm, be the iterated order of a meromorphic function \(f\).NEWLINENEWLINEUsing methods of Nevanlinna theory the authors prove some new results on the growth of the logarithmic derivative.NEWLINENEWLINETheorem 1.1. Suppose that \(k\geq 2\) is an integer, and let \(f\) be a meromorphic function. Then NEWLINE\[NEWLINE \rho_1\Bigl(\frac{f'}{f}\Bigr)=\max\Bigl\{ \rho_1\Bigl(\frac{f^{(k)}}{f}\Bigr) : k\geq 2\Bigr\} .NEWLINE\]NEWLINENEWLINENEWLINENEWLINE Theorem 1.2. Let \(f\) be an entire function with a finite number of zeros. Then for any integer \(k\geq 1\) NEWLINE\[NEWLINE \rho_1\Bigl (\frac{f^{(k)}}{f}\Bigr)= \rho_2(f) .NEWLINE\]NEWLINENEWLINENEWLINEThese theorems are applied to the investigation of linear differential equations.